This Only Digit Carries The Most Information

Relevant wiki: Arithmetic Puzzles - Fill in the Blanks

The long multiplication tells us that we want to find 2 single digit positive integers such that their product is a 2-digit integer with 7 as its tenth digit.

So the possible value of this product can take the value $70, 71, 72, \ldots, 79$ .

The product cannot be 70 because $70=7\times10$ cannot be expressed as the product of 2 single digit positive integers. Likewise, we can rule out 74, 75, 77 and 78 as well.

The product cannot be 71 because $71 = 1\times71$ is a prime number, and so it cannot be expressed as the product of 2 single digit positive integers. Likewise, we can rule out 73 and 79 as well.

The product can be $72=8\times9$ because it can be expressed as product of 2 single digit positive integers.

Hence, the long multiplcation is

$\Large{\begin{array}{cccc} & & & &\boxed8 \\ \times & & & &\boxed9 \\ \hline & & & 7 &\boxed2 \\ \hline \end{array}}$

with the positions of the digits 8 and 9 to be interchangeable. Thus, our answer is $8\times9\times2 = 72\times2=\boxed{144}$ .

$\Large{\begin{array}{cccc} & & & &8\\ \times & & & &9\\ \hline & & & 7 & 2\\ \hline \end{array}}$

$\Rightarrow 8 \times 9 \times 2 = \boxed{144}$

Since they are all single digits, the only possibility is 9x8=72. Other answers you may have thought of was 7x10, 7x11 etc. but a quick realisation is that they are all single digits.